MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Physics
Department
8.323:
Relativistic
Quantum
Field
Theory
I
Prof. Alan
Guth
April
2,
2008
LECTURE
NOTES
6
PATH
INTEGRALS,
GREEN’S
FUNCTIONS,
AND
GENERATING
FUNCTIONALS
In
these
notes
we
will
extend
the
path
integral
methods
discussed
in
Lecture
Notes
5
to
describe
Green’s
functions,
which
we
define
to
be
ground
state
expecta
tion
values
of
the
timeordered
product
of
Heisenberg
operators. For
the
case
of
a
nonrelativistic
particle
moving
in
one
dimension,
discussed
in
Lecture
Notes
5,
the
Green’s
functions
can
be
written
as
G
(
t
N
,
. . . ,
t
1
)
≡
0

T
{
x
(
t
N
)
x
(
t
N
−
1
)
. . .
x
(
t
1
)
}
0
(6.1)
=
0

x
(
t
N
)
x
(
t
N
−
1
)
. . .
x
(
t
1
)

0
,
where

0
denotes
the
ground
state,
and
the
second
line
assumes
that
we
have
labeled
the
time
arguments
so
that
they
are
timeordered,
in
the
sense
that
t
N
≥
t
N
−
1
≥
. . .
≥
t
1
.
(6.2)
In
the
quantum
field
theory,
the
Green’s
functions
will
be
defined
analogously
by
G
(
x
N
,
. . . ,
x
1
)
≡
0

T
{
φ
(
x
N
)
. . .
φ
(
x
1
)
}
0
,
(6.3)
=
0

φ
(
x
N
)
. . .
φ
(
x
1
)

0
,
where

0
denotes
the
vacuum
state,
and
again
the
second
line
assumes
that
the
time
arguments
are
timeordered. In
the
nonrelativistic
quantum
mechanics
example
of
Eq. (6.1),
the
Green’s
functions
are
not
quantities
that
are
particularly
interesting,
so
they
are
usually
never
mentioned
in
a
course
in
quantum
mechanics.
We
will
soon
see,
however,
that
the
quantum
field
theory
Green’s
functions
of
Eq. (6.3)
are
very
interesting. In
particular,
the
entire
formalism
for
calculating
scattering
cross
sections
and
decay
rates
will
be
based
upon
relating
these
quantities
to
the
Green’s
functions. In
addition
to
showing
how
to
express
these
Green’s
functions
as
path
integrals,
in
these
notes
we
will
also
see
that
one
can
define
a
generating
functional
Z
[
J
]
in
such
a
way
that
the
Green’s
functions
can
be
expressed
simply
in
terms
of
the
functional
derivatives
of
the
generating
functional.